Relative K0, annihilators, Fitting ideals and the Stickelberger phenomena

36 Pages·2012·0.25 MB·English

Relative K , annihilators, Fitting ideals and
0
the Stickelberger phenomena
Victor P. Snaith
Abstract
When G is abelian and l is a prime we show how elements of the
relative K-group K (Z [G];Q ) give rise to annihilator/Fitting ideal
0 l l
relations of certain associated Z[G]-modules. Examples of this phe-
nomenon are ubiquitous. Particularly, we give examples in which G is
the Galois group of an extension of global (cid:12)elds and the resulting an-
nihilator/Fitting ideal relation is closely connected to Stickelberger’s
Theorem and to the conjectures Coates-Sinnott and Brumer.
1 Introduction
1.1 Suppose thatl isaprimeandGisa(cid:12)nitegroup. Inthiscase therelative
K-group K (Z [G];Q ) appearing in the localisation sequence (see 2.1) is
0 l l
x
isomorphic to the zero-th K-group of the category of (cid:12)nite Z [G]-modules
l
of (cid:12)nite projective dimension, usually denoted by K T(Z [G]). When G is
0 l
abelian we have an isomorphism of the form K (Z [G];Q ) = Ql[G](cid:3).
0 l l (cid:24) Zl[G](cid:3)
As explained in 2, elements of K (Z [G];Q ) may be constructed from a
0 l l
x
boundedperfectcomplexofZ [G]-modulestogetherwithaQ [G]-trivialisation
l l
of its cohomology. In arithmetic and algebraic geometry there are many sit-
uations, some of which are studied in 3-5, which give rise to this data. In
xx
this paper we shall consider the simplest example of this type of element,
namely the case of a perfect complex of Z [G]-modules whose cohomology
l
groups are (cid:12)nite. Our main result concerns the simplest case of all, when the
complex has only two non-zero, (cid:12)nite cohomology groups. In this case (The-
orem 2.4) when G is abelian, there are Stickelberger-type relations between
the element of Q [G](cid:3) which represents the relative K class, the annihilators
l 0
and the Fitting ideals of the cohomology groups. In order to apply Theorem
2.4 one must evaluate evaluate the element of Q [G](cid:3) representing the class
l
on K (Z [G];Q ). In general this is di(cid:14)cult to do but in the cases of abelian
0 l l
Galois extensions of function (cid:12)elds and totally real number (cid:12)elds this element
turns out to be a classical higher order Stickelberger element (see Theorems
1.6 and 3.4).
1
Let us begin by recalling the higher Stickelberger elements and the con-
jectures pertaining to them. Suppose that L=Q is a (cid:12)nite Galois extension of
number (cid:12)elds with abelian Galois group, G(L=Q), and L totally real. Then,
for each integer n 2, there is a unique unit of the rational group-ring
(cid:21)
(cid:2) (n) Q[G(L=Q)](cid:3)
L=Q
2
such that
(cid:31)((cid:2) (n)) = L (1 n;(cid:31)(cid:0)1)
L=Q Q
(cid:0)
for each one-dimensional complex representation (cid:31) where L (s;(cid:31)(cid:0)1) denotes
Q
the Dirichlet L-function of the character (cid:31)(cid:0)1 ([58] Ch.4). The rationality of
(cid:2) (n) is seen by writing the L-function in terms of partial zeta functions
L=Q
L (1 n;(cid:31)(cid:0)1) = (cid:31)(g)(cid:0)1(cid:16) (g;1 n)
Q (cid:0) X Q (cid:0)
g2G(LKer((cid:31))=Q)
and recalling that (cid:16) (g;1 n) is a rational number, by a result of Klingen
Q
(cid:0)
and Siegel (cf. [48]).
Inspired by Stickelberger’s Theorem ([58] p.94), Coates and Sinnott made
afundamentalconjecture concerningGaloisactionsonthealgebraicK-groups
of algebraic integers in number (cid:12)elds.
Conjecture 1.2 ([11]; see also [12]).)
In the situation of 1.1, let l be a prime and let denote the ring of
L
x O
algebraic integers of L. For each n 1
(cid:21)
w (Q)(cid:2) (n+1) ann (H0(Spec( );Q =Z (n+1)))
n+1 L=Q (cid:1) Zl[G(L=Q)] (cid:19)et OL l l
ann (K ( ) Z ):
(cid:18) Zl[G(L=Q)] 2n OL (cid:10) l
Here w (Q) denotes the largest integer m such that the Galois group,
n+1
G(Q(e2(cid:25)i=m)=Q), has exponent dividing n+ 1. Alternatively it is the order
of H0(Spec(Z);Q=Z(n+1)) (see ([50] 7.2.4).
e(cid:19)t x
Conjecture 1.3 In the situation of 1.1 and Conjecture 1.2 one could also
x
conjecture the stronger result that, for each n 1,
(cid:21)
(cid:2) (n+1) ann (H0(Spec( );Q =Z (n+1)))
L=Q (cid:1) Zl[G(L=Q)] (cid:19)et OL l l
F (K ( ) Z ):
(cid:18) Zl[G(L=Q)] 2n OL (cid:10) l
Here F (M) denotes the Fitting ideal of the module M (see 2).
Zl[G(L=Q)] x
This conjecture is stronger than Conjecture 1.2 since the integer w (Q)
n+1
has been removed and the Fitting ideal is contained in the annihilator ideal
(see 2.3).
x
2
Remark 1.4 (i) The classical theorem of Stickelberger ([58] p.94) corre-
sponds to the case when n = 0 in Conjecture 1.2 when K ( ) is replaced
0 L
O
by its torsion subgroup, the ideal class group. The Brumer Conjecture ([4],
[60]), which is still open, corresponds to the case when n = 0 and L=K is an
abelian extension with K totally real.
(ii) Furthermore, inspired by the Brumer Conjecture, a conjecture similar
to Conjecture 1.3 makes sense for any abelian Galois extension of number
(cid:12)elds, L=K, with L totally real (see [50] Conjecture 7.2.6).
(iii) There is an equivariant Chern class homomorphism of the form (cf.
[2] Theorem B; [14])
c : K ( ) Z H1(Spec( [1=l]);Q =Z (n+1)):
n+1;2 2n OL (cid:10) l (cid:0)! (cid:19)et OL l l
The Lichtenbaum-Quillen Conjecture predicts (at least when l is odd) that
c is an isomorphism. This was proved for n = 1 in [55]. As a corollary of
n+1;2
the fundamental results of Voevodsky [56] [57], when l = 2 this Chern class is
nearly an isomorphism in all dimensions [43]. Voevodsky’s method requires
the existence of suitable \norm varieties" which is not yet established for all
odd primes. However, recent work by Rost combined with that of Suslin-
Voevodsky shows that c is an isomorphism for l = 3.
n+1;2
The annihilator
ann (H0(Spec( );Q =Z (n+1)))
Zl[G(L=Q)] (cid:19)et OL l l
is well-known ([10] Lemma 2.3; [54] p.82). Explicitly, for n 0, this annihi-
(cid:21)
lator is equal to the ideal generated by the elements, (P;L=Q) NPn+1
(cid:0) 2
Z [G(L=Q)], where P runs through primes which ramify in L=Q or divide
l
w (L) = H0(Spec( );Q =Z (n+1)) :
n+1 j (cid:19)et OL l l j
Here (P;L=Q) G(L=Q) is the element corresponding to P under Artin
2
reciprocity.
In[14]itisshownthat(theresultcanalsobededucedfrom[15])K ( )
2n L
O (cid:10)
Z mapssurjectivelyontoH1(Spec( [1=l]);Q =Z (n+1))sothattheCoates-
l (cid:19)et OL l l
SinnottConjectureof 1.2,asmodi(cid:12)edinConjecture1.3,predictsaninclusion
x
of the form
(cid:2) (n+1) ann (H0(Spec( );Q =Z (n+1)))
L=Q (cid:1) Zl[G(L=Q)] (cid:19)et OL l l
F (H1(Spec( [1=l]);Q =Z (n+1))):
(cid:18) Zl[G(L=Q)] (cid:19)et OL l l
(iv) The annihilator/Fittingidealrelationof(iii)makes sense moregener-
ally. Let L=K be a Galois extension of totally real number (cid:12)elds with abelian
Galois group, G(L=K). Let S be a (cid:12)nite set of primes of K including all
3
those which ramify in L=K. Then, as in 1.1, for each n 2 there exists a
x (cid:21)
unique unit of the rational group-ring
(cid:2) (n) Q[G(L=K)](cid:3)
L=K;S
2
such that
(cid:31)((cid:2) (n)) = L (1 n;(cid:31)(cid:0)1)
L=K;S K;S
(cid:0)
for each one-dimensional complex representation, (cid:31). Here (cid:31) is a character of
G(L=K) and L (s;(cid:31)(cid:0)1) denotes the Artin L-function of (cid:31)(cid:0)1 with the Euler
K;S
factors associated to the primes in S removed.
Let S0 be the set of places of L over S. Set Xl = Spec(OL;S0[1=l]), where
OL;S0 is the ring of S0-integers of L. For each n 1, one would expect an
(cid:21)
annihilator relation of the form
(cid:2) (n+1) ann (H0(Spec( );Q =Z (n+1)))
L=K;S (cid:1) Zl[G(L=K)] (cid:19)et OL l l
F (H1(X ;Q =Z (n+1)))(cid:1)Z [G(L=K)]:
(cid:18) Zl[G(L=K)] e(cid:19)t l l l l
In fact, it can be shown that these ideals have the same radical ([50] Theorem
7.3.2).
(v) Using Iwasawa theory, it is shown in ([11] Theorem 2) that if L=Q is
a totally real abelian extension of conductor f, l is a prime and b is an integer
coprime to fl then w (Q)(b2 (b;L=Q))(cid:2) (2) Q [G(L=Q)] lies in
2 L=Q l
(cid:0) 2
ann (H1(Spec( );Q =Z (2))) = ann (K ( ) Z ):
Zl[G(L=K)] (cid:19)et OL l l Zl[G(L=K)] 2 OL (cid:10) l
By combining ([11] Theorem 2.1) with results from [46], it is shown
more generally in [39], under the same conditions (actually the condition
that (b;fl) = 1 seems to have been omitted in ([39] Th(cid:19)eor(cid:18)eme 2.2)), that
w (Q)(bj+1 (b;L=Q))(cid:2) (j +1) Q [G(L=Q)] lies in
j+1 L=Q l
(cid:0) 2
ann (H1(Spec( );Q =Z (j +1)))
Zl[G(L=K)] (cid:19)et OL l l
for j = 1;3;5:::.
1.5 Now let us describe the main results of this paper. Let l be an odd
prime and let m be a positive integer prime to l. If (cid:24) denotes the root
t
of unity (cid:24) = e2(cid:25)i=t, consider the cyclotomic (cid:12)eld Q((cid:24) ) and its totally
t mls+1
real sub(cid:12)eld Q((cid:24) )+. Let S denote the primes of Q((cid:24) )+ which
mls+1 ml mls+1
divide ml and let X+ = Spec(Z[(cid:24) ]+ ) denote the spectrum of the S -
l mls+1 Sml ml
integers in Q((cid:24) )+. Note that H0(X+;Q =Z (1 r)) is isomorphic to
mls+1 (cid:19)et l l l (cid:0)
H0(Z[(cid:24) ][1=ml];Q =Z (1 r)) for all negative odd integers r.
(cid:19)et mls+1 l l (cid:0)
The following result will be proved in 4.3.
x
4
Theorem 1.6
Let l be an odd prime. Let l;m and X+ be as in the notation of 1.5 and
l x
let (cid:2) (1 r) denote the higher Stickelberger element of 1.1. Then,
Q((cid:24)mls+1)+=Q (cid:0) x
for r = 1; 3; 5;::: and any positive integer s:
(cid:0) (cid:0) (cid:0)
(i) there exists a chain of annihilator ideal relations for (cid:19)etale cohomology
of the form
ftm0 j t 2 annZl[G(Q((cid:24)mls+1)+=Q)](H(cid:19)e1t(Xl+;Ql=Zl(1(cid:0)r)))g
(cid:2) (1 r)ann (H0(X+;Q =Z (1 r)))
(cid:18) Q((cid:24)mls+1)+=Q (cid:0) Zl[G(Q((cid:24)mls+1)+=Q)] (cid:19)et l l l (cid:0)
ann (H1(X+;Q =Z (1 r))):
(cid:18) Zl[G(Q((cid:24)mls+1)+=Q)] (cid:19)et l l l (cid:0)
(ii) If l does not divide m 1 then in (i) the (cid:12)nal
(cid:0)
ann (H1(X+;Q =Z (1 r)))
Zl[G(Q((cid:24)mls+1)+=Q)] (cid:19)et l l l (cid:0)
may be replaced by
F (H1(X+;Q =Z (1 r))):
Zl[G(Q((cid:24)mls+1)+=Q)] (cid:19)et l l l (cid:0)
Here m is the minimal number of generators of the Z [G(Q((cid:24) )+=Q)]-
0 l mls+1
module H1(X+;Q =Z (1 r)) and F (M) denotes the Fitting
(cid:19)et l l l (cid:0) Zl[G(Q((cid:24)mls+1)+=Q)]
ideal of M.
From the localisation sequence of ([52] p.268), when r = 1; 3; 5;:::,
(cid:0) (cid:0) (cid:0)
H1(Spec(Z[(cid:24) ]+[1=l]);Q =Z (1 r))isaGaloissubmoduleofH1(X+;Q =Z (1
(cid:19)et mls+1 l l (cid:0) (cid:19)et l l l (cid:0)
r)), which implies the following result.
Corollary 1.7
In the notation of 1.5 and Theorem 1.6, when l is an odd prime not
x
dividing m and r = 1; 3; 5;:::
(cid:0) (cid:0) (cid:0)
(cid:2) (1 r)ann (H0(Spec(Z[(cid:24) ]+[1=l]);Q =Z (1 r)))
Q((cid:24)mls+1)+=Q (cid:0) Zl[G(Q((cid:24)mls+1)+=Q)] (cid:19)et mls+1 l l (cid:0)
ann (H1(Spec(Z[(cid:24) ]+[1=l];Q =Z (1 r)))
(cid:18) Zl[G(Q((cid:24)mls+1)+=Q)] (cid:19)et mls+1 l l (cid:0)
for all positive integers s.
The paper is organised in the following manner. In 2 we introduce the
x
relative K-group K (Z [G];Q ) for any (cid:12)nite group G and explain how ele-
0 l l
ments of this group can yield annihilator/Fitting ideal relations (Theorem
2.4). Two important classes of elements to which the construction of 2 ap-
x
pliesariseinconnectionwiththe(cid:19)etalecohomologygroupsofcurves over(cid:12)nite
(cid:12)elds and rings of integers in number (cid:12)elds. In 3 we apply Theorem 2.4 to
x
5
an example from ([8] 7) to prove a Coates-Sinnott type of result (Theorem
x
3.4) for the (cid:19)etale cohomology groups of curves over (cid:12)nite (cid:12)elds. In 4 we use
x
an example from [5] to prove a similar result (Theorem 1.6 proved in 4.3) for
x
the (cid:19)etale cohomology H1(X+;Q =Z (1 r)) of the S-integers in the totally
(cid:19)et l l l (cid:0)
real sub(cid:12)eld of a cyclotomic (cid:12)eld. When r = 2; 4; 6;::: { the case not
(cid:0) (cid:0) (cid:0)
covered by Theorem 1.6 { we apply Theorem 2.4 in Theorem 4.6 to study
the annihilator ideal of H2(X+;Z (1 r)). In 4.8 and Theorem 4.9 we use
(cid:19)et l l (cid:0) x
Theorem 4.6 together with calculation of Beilinson [3] to construct elements
of this annihilator from the leading terms of the Dirichlet L-function at s = r.
In 5 we discuss invariants lying in K (Z [G];Q ) which are constructed from
0 l l
x
relatively abelian extensions of totally real number (cid:12)elds with vanishing Iwa-
sawa (cid:22)-invariants, from the Galois action on vanishing cycles and from (cid:19)etale
coverings of curves and surfaces.
Finally, a remark about the classical Stickelberger Theorem and Brumer
Conjecture which, as mentioned in Remark 1.4, correspond to the case when
r = 0. In ([50] Remark 7.2.11(ii)) it is explained how one might use \Tate
sequence" constructed in [41] to approach the Brumer Conjecture by the
method used to prove Theorem 1.6. This approach is much more di(cid:14)cult
thantheproofofTheorem1.6becauseofthepresenceofnon-trivialregulators
(cf. [50] 7.1.12). On the other hand, this approach would aim to prove a
x
statement(analogoustothatofTheorem1.6)statingthatatoddprimesl,the
Stickelberger ideal lies in the annihilator (and sometimes the Fitting ) ideal
of the class-group of the S-integers (where S-contains all rami(cid:12)ed primes).
Even in the case of the classical Stickelberger Theorem this seems to lead to
questions which are at present unanswered. In the classical Stickelberger case
such a Fitting ideal inclusion fails for l = 2 but holds at odd primes when,
for example, both the roots of unity and the class-group are cohomologically
trivial.
Very recently I became aware of the very important results of Masato
Kurihara [27] which show that the Fitting ideal appearing in Theorem 1.6(ii)
is given by a generalised Stickelberger ideal. Using Kurihara’s results one
may verify the second inclusion in the statements of Theorems 1.6(ii) without
any restrictions on the integer m while the other inclusion may in turn be
combined with the results of [27] to yield upper bounds on the number of
Z [G]-module generators of the (cid:19)etale cohomology groups.
l
Ackowledgments The results of [5] are crucial to this paper and I am
particularly indebted to David Burns for bringing his work to my attention.
ThisworkwasbegunduringavisittotheUniversity ofPennsylvaniainMarch
2000 and completed during a visit, funded by a Special Project Grant from
the Royal Society, to the Euler Institute in St Petersburg during September
2000. I am particularly grateful to these institutions for their hospitality
and to mathematicians there { Ted Chinburg, Ivan Panin and Igor Zhukov
6
{ for help and encouragement. In addition, conversations with John Coates,
Cornelius Greither, Bernhard Ko(cid:127)ck, Masato Kurihara, Christian Popescu,
Nguyen Thong Quang Do, Ju(cid:127)rgen Ritter, Al Weiss and Andrew Wiles have
been very helpful. In fact, [11] was one of the papers which (cid:12)rst got me
interested in algebraic K-theory a long time ago.
2 K (Z [G];Q ) and annihilator/Fitting ideal
0 l l
relations
2.1 Let l be a prime, G a (cid:12)nite group and let f : Z [G] Q [G] denote the
l l
(cid:0)!
homomorphism of group-rings induced by the inclusion of the l-adic integers
intothefraction(cid:12)eld, thel-adicrationals. WriteK (Z [G];Q )fortherelative
0 l l
K-group of f, denoted by K (Z [G];f) in ([53] p.214; see also [50] De(cid:12)nition
0 l
2.1.5). By ([53] Lemma 15.6) elements of K (Z [G];Q ) are represented by
0 l l
triples [A;g;B] where A;B are (cid:12)nitely generated, projective Z [G]-modules
l
and g is a Q [G]-module isomorphism of the form g : A Q (cid:24)= B Q .
l (cid:10)Zl l ! (cid:10)Zl l
De(cid:12)ning an exact sequence of triples in the obvious manner, the relations
between these elements are generated by the following two types:
(i) [A;g;B] = [A0;g0;B0]+[A00;g00;B00] if there exists an exact sequence
0 (A0;g0;B0) (A;g;B) (A00;g00;B00) 0
(cid:0)! (cid:0)! (cid:0)! (cid:0)!
and
(ii) [A;gh;C] = [A;h;B]+[B;g;C].
This group (cid:12)ts into a localisation sequence of the form ([44] 5 Theorem
x
5; [20] p.233)
f(cid:3) @ (cid:25) f(cid:3)
K (Z [G]) K (Q [G]) K (Z [G];Q ) K (Z [G]) K (Q [G]):
1 l 1 l 0 l l 0 l 0 l
(cid:0)! (cid:0)! (cid:0)! (cid:0)!
Assume now that G is abelian. In this case K (Q [G]) = Q [G](cid:3) because
1 l (cid:24) l
Q [G] is a product of (cid:12)elds and K (Z [G]) = Z [G](cid:3) ([13]I p.179 Theorem
l 1 l (cid:24) l
(46.24)). Under these isomorphisms f is identi(cid:12)ed with the canonical inclu-
(cid:3)
sion.
The homomorphism, K (Z [G]) f(cid:3) K (Q [G]), is injective for all (cid:12)nite
0 l 0 l
(cid:0)!
groups G ([47] Theorem 34 p.131; [13]II p.47 Theorem 39.10). Alternatively,
whenGisabelian, Z [G]issemi-localandtheinjectivity off followsfromthe
l (cid:3)
fact that a (cid:12)nitely generated projective module over a local ring is free ([16]
p.124 Corollary 4.8 and p.205 Exercise 7.2). Thus the localisation sequence
yields an isomorphism of the form
Q [G](cid:3)
l
K (Z [G];Q ) =
0 l l (cid:24) Z [G](cid:3)
l
7
when G is abelian. From the explicit description of @ ([53] p.216) this iso-
morphism sends the coset of (cid:11) Q [G](cid:3) to [Z [G];((cid:11) );Z [G]]. The inverse
l l l
2 (cid:1)(cid:0)
isomorphism sends [A;g;B], where A and B may be assumed to be free
Z [G]-modules, to the coset of det(g) Q [G](cid:3) with respect to any choice of
l l
2
Z [G]-bases for A and B.
l
Example 2.2 We shall be particularly interested in the following source of
elements of K (Z [G];Q ).
0 l l
As in 2.1, let l be a prime and let G be a (cid:12)nite abelian group. Suppose
x
that
0 F dk F dk(cid:0)1 ::: d2 F d1 F 0
k k(cid:0)1 1 0
(cid:0)! (cid:0)! (cid:0)! (cid:0)! (cid:0)! (cid:0)!
is a bounded complex of (cid:12)nitely generated, projective Z [G]-modules (i.e. a
l
perfect complex of Z [G]-modules), having all its homology groups (cid:12)nite.
l
As usual, let Z = Ker(d : F F ) and B = d (F ) F denote
t t t t(cid:0)1 t t+1 t+1 t
(cid:0)! (cid:18)
the Z [G]-modules of t-dimensional cycles and boundaries, respectively. We
l
have short exact sequences of the form
0 B (cid:30)i Z H (F ) 0
i i i (cid:3)
(cid:0)! (cid:0)! (cid:0)! (cid:0)!
and
0 Z i+1 F di+1 B 0:
i+1 i+1 i
(cid:0)! (cid:0)! (cid:0)! (cid:0)!
Applying ( Q ) we obtain isomorphisms
l
(cid:0)(cid:10)
(cid:30) : B Q (cid:24)= Z Q
i i l i l
(cid:10) (cid:0)! (cid:10)
and we may choose Q [G]-module splittings of the form
l
(cid:17) : B Q F Q
i i l i+1 l
(cid:10) (cid:0)! (cid:10)
such that (d 1)(cid:17) = 1 : B Q B Q .
i+1 i i l i l
(cid:10) (cid:10) (cid:0)! (cid:10)
Then we form a Q [G]-module isomorphism of the form
l
X : F Q (cid:24)= F Q
j 2j l j 2j+1 l
(cid:8) (cid:10) (cid:0)! (cid:8) (cid:10)
given by the composition
F Q (cid:8)j ( 2j+(cid:17)2j(cid:0)1)(cid:0)1 (Z Q ) (B Q )
j 2j l j 2j l 2j(cid:0)1 l
(cid:8) (cid:10) (cid:0)! (cid:8) (cid:10) (cid:8) (cid:10)
(cid:8)j ((cid:30)(cid:0)2j1(cid:8)(cid:30)2j(cid:0)1) (B Q ) (Z Q )
j 2j l 2j(cid:0)1 l
(cid:0)! (cid:8) (cid:10) (cid:8) (cid:10)
((cid:8)j (cid:17)2j+ 2j(cid:0)1) F Q :
j 2j(cid:0)1 l
(cid:0)! (cid:8) (cid:10)
8
This construction de(cid:12)nes a class, [ F ;X; F ], in K (Z [G];Q ) which
j 2j j 2j+1 0 l l
(cid:8) (cid:8)
is well-known to be independent of the choices of the splittings used to de(cid:12)ne
X ([53] Ch. 15; see also [50] Propositions 2.5.35 and 7.1.8).
We shall denote by
Q [G](cid:3)
l
det(X)
2 Z [G](cid:3)
l
the element which corresponds to [ F ;X; F ] K (Z [G];Q ) under
j 2j j 2j+1 0 l l
(cid:8) (cid:8) 2
the isomorphism of 2.1.
x
2.3 Recall that, if R is aring and M a (left)R-module, the (left) annihilator
ideal ann (M)(cid:1)R of M is de(cid:12)ned to be
R
ann (M) = r R r m = 0 for all m M :
R
f 2 j (cid:1) 2 g
Let us recall from ([35] Appendix; see also [59]) the properties of the
Fitting ideal (referred to as the initial Fitting invariant in [40]).
Let R be a commutative ring with identity and let M be a (cid:12)nitely pre-
sented R-module, in our applications M will actually be (cid:12)nite. Suppose that
M has a presentation of the form
Ra f Rb M 0
(cid:0)! (cid:0)! (cid:0)!
with a b then the Fitting ideal of the R-module M, denoted by F (M), is
R
(cid:21)
the ideal of R generated by all b b minors of any matrix representing f.
(cid:2)
The Fitting ideal F (M) is independent of the presentation chosen for M
R
and is contained in the annihilator ideal of M, F (M) ann (M). If M is
R R
(cid:18)
generated by n elements then ann (M)n F (M) and if (cid:25) : M M0 is a
R R
(cid:18) !
surjection of (cid:12)nitely presented R-modules then F (M) F (M0).
R R
(cid:18)
The following result yields relations between the annihilator ideals and
Fitting ideals of the homology modules in Example 2.2 in the special case
when each H (F ) is (cid:12)nite and zero except for i = 0;1.
i (cid:3)
Theorem 2.4 Let G be a (cid:12)nite abelian group and l a prime. Suppose that
0 F dk F dk(cid:0)1 ::: d2 F d1 F 0
k k(cid:0)1 1 0
(cid:0)! (cid:0)! (cid:0)! (cid:0)! (cid:0)! (cid:0)!
is a bounded, perfect complex of Z [G]-modules, as in Example 2.2, having
l
H (F ) (cid:12)nite for i = 0;1 and zero otherwise. Let
i (cid:3)
Q [G](cid:3)
l
[ F ;X; F ] K (Z [G];Q ) =
(cid:8)j 2j (cid:8)j 2j+1 2 0 l l (cid:24) Z [G](cid:3)
l
be as in Example 2.2. Then:
9
(i) if t ann (H (F )),
i 2 Zl[G] i (cid:3)
det(X)((cid:0)1)itmi ann (H (F ))(cid:1)Z [G]
i 2 Zl[G] 1(cid:0)i (cid:3) l
for i = 0;1. Here m ;m is the minimal number of generators required for
0 1
the Z [G]-module H (F );Hom(H (F );Q =Z ), respectively,
l 0 (cid:3) 1 (cid:3) l l
(ii) if the Sylow l-subgroup of G is cyclic then in (i) ann (H (F ))
Zl[G] 1(cid:0)i (cid:3)
may be replaced by F (H (F )).
Zl[G] 1(cid:0)i (cid:3)
2.5 The proof of Theorem 2.4 will occupy the remainder of this section,
culminating in 2.11. We begin with some preliminary observations and con-
x
structions.
In the situation of Theorem 2.4, without changing the class
[ F ;X; F ], we may replace F ;:::;F by (cid:12)nitely generated, free
j 2j j 2j+1 0 k(cid:0)1
(cid:8) (cid:8)
Z [G]-modules. Then, since ( 1)i[F ] = f ((cid:25)([ F ;X; F ])) = 0
l Pi (cid:0) i (cid:3) (cid:8)j 2j (cid:8)j 2j+1
and f is injective, F must be stably free and hence free. Therefore hence-
(cid:3) k
forth we shall assume that each F is a (cid:12)nitely generated, free Z [G]-module.
i l
In this case we are going to modify the isomorphism X to make it Z -integral.
l
Firstly we observe that, for i 1, the short exact sequence of Z [G]-
l
(cid:21)
module homomorphisms
0 B i+1 F di+1 B 0
i+1 i+1 i
(cid:0)! (cid:0)! (cid:0)! (cid:0)!
splits. To see this, write G = H G where G is the Sylow l-subgroup of G
1 1
(cid:2)
and observe, by induction starting with B = F , that B is a torsion-free,
k(cid:0)1 (cid:24) k i
cohomologically trivial Z [G ]-module if i 1. Hence, by ([1] Theorem 7)
l 1
(cid:21)
there exist elements v ;:::;v B such that the natural map induces an
1 t i
2
isomorphism of the form
Z=lm[G ] < v ;:::;v > (cid:24)= B =lmB
1 1 t i i
(cid:0)!
for each m 1. Since B is l-adically complete, the inverse limit of these
i
(cid:21)
isomorphisms shows that B is a free Z [G ]-module. Hence we may split d
i l 1 i+1
as a Z [G ]-module homomorphism and, by averaging over H, as a Z [G]-
l 1 l
module homomorphism also.
Therefore we have
(cid:17) : B F
i i i+1
(cid:0)!
for i 1 and
(cid:21) =
(cid:30) : B Z
i i i
(cid:0)!
for i 2 without tensoring with the l-adic rationals.
(cid:21)
Since H (F ) is (cid:12)nite, in the notation of Example 2.2, we have a homo-
0 (cid:3)
morphism
(cid:17) = (cid:17) ((cid:30) 1)(cid:0)1 : F Q F Q
0 0 0 l 1 l
(cid:1) (cid:10) (cid:10) (cid:0)! (cid:10)
10